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2. Properties of Real Numbers
1.4
Properties of Real Numbers
Use the distributive property.
Use the inverse properties.
Use the identity properties.
Use the commutative and associative properties.
Use the multiplication property of 0. 1 2 3 4 5
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Objective 1
Use the distributive property.
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The Distributive Property
For any real numbers, a, b, and c,
a(b + c) = ab + ac and (b + c)a = ba + ca.
The Distributive property can be extended to more than two numbers.
a(b + c + d) = ab + ac + ad
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EXAMPLE 1
Use the distributive property to rewrite each expression.
a. 4(p – 5)
= 4(p – 5) =
= 4p + 20
b. 6m + 2m
= 6m + 2m =
= 4m
4p
(4)(5)
(6 + 2)m
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continued
c. 2r + 3s
Because there is no common number or variable here, we cannot use the distributive property to rewrite the expression.
d. 5(4p – 2q + r)
= 5(4p – 2q + r) =
= 20p – 10q + 5r
5(4p)
5(2q)
+ 5r
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Objective 2
Use the inverse properties.
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Inverse Properties
For any real number a,
a + a = and a + a = 0
and
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Objective 3
Use the identity properties.
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The numbers 0 and 1 each have a special property. Zero is the only number that can be added to any number to get that number.
Adding 0 to any number leaves the identity of the number unchanged. For this reason, 0 is called the identity element for addition, or the additive identity.
In a similar way, multiplying any number by 1 leaves the identity of the number unchanged, so 1 is the identity element for multiplication, or the multiplicative identity.
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EXAMPLE 2
Simplify each expression.
a. x – 3x = 1x – 3x
= 1x – 3x Identity property.
= (1 – 3)x Distributive property.
= 2x Subtract inside parentheses.
b. (3 + 4p) = 1(3 + 4p)
= 1(3) + (1)(4p) Identity property.
= 3 – 4p Multiply.
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A term is a number or the product of a number and one or more variables raised to powers.
The numerical factor in a term is called the numerical coefficient, or just the coefficient.
Terms with exactly the same variables raised to exactly the same powers are called like terms.
Examples: 5y and 21y 6x2 and 9x2
Some examples of unlike terms are
3m and 16x y3 and 3y2
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Objective 4
Use the commutative and associative properties.
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Commutative and Associative Properties
For any real numbers a, b, and c,
a + b = b + a
ab = ba
and
Interchange the order of the two terms or factors.
Also, a + (b + c) = (a + b) + c
and a(bc) = (ab)c.
Shift parentheses among the three terms or factors; the order stays the same.
Commutative properties
Associative properties
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EXAMPLE 3
Simplify 12b – 9 + 4b – 7b + 1.
12b – 9 + 4b – 7b + 1
= (12b + 4b) – 9 – 7b + 1
= (12 + 4)b – 9 – 7b + 1
= 16b – 9 – 7b + 1
= (16b – 7b) – 9 + 1
= (16 – 7)b – 9 + 1
= 9b – 8
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EXAMPLE 4
Simplify each expression.
a. 12b – 9b + 5b – 7b
= (12 – – 7)b Distributive property.
= b Combine like terms.
b. 6 – (2x + 7) – 3
= 6 – 2x – 7 – 3 Distributive property.
= –2x + 6 – 7 – 3 Commutative property.
= –2x – 4 Combine like terms.
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continued
Simplify each expression.
c. 4m(2n)
= (4)(2)mn
= 8mn
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Objective 5
Use the multiplication property of 0.
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Multiplication Property of 0
For any real number a,
a  0 = and 0  a = 0.
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